Highly sensitive thermal conductivity measurements of suspended membranes (SiN and diamond) using a 3w-Volklein method

0.92 MB·English

by  A. Sikora

Highly sensitive thermal conductivity measurements of suspended
membranes (SiN and diamond) using a 3ω-Vo¨lklein method
A. Sikora,1 H. Ftouni,1 J. Richard,1 C. H´ebert,1 D. Eon,1 F. Omn`es,1 and O. Bourgeois1,∗
2 1Institut NE´EL, CNRS-UJF, 25 avenue des Martyrs, 38042 Grenoble Cedex 9, France
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(Dated: January 20, 2012)
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Abstract
a
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9 Asuspendedsystem for measuringthe thermalproperties of membranes is presented. Thesensi-
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tivethermalmeasurementisbasedonthe3ω dynamicmethodcoupledtoaVo¨lkleingeometry. The

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l device obtained using micro-machining processes allows the measurement of the in-plane thermal
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h conductivityofamembranewithasensitivityoflessthan10nW/K(+/-5 10−3Wm−1K−1atroom
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×
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e temperature) and a very high resolution (∆K/K = 10−3). A transducer (heater/thermometer)
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. centered on the membrane is used to create an oscillation of the heat flux and to measure the
t
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m temperature oscillation at the third harmonic using a Wheatstone bridge set-up. Power as low as
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d 0.1nanoWatt has been measured at room temperature. The method has been applied to measure
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thermal properties of low stress silicon nitride and polycrystalline diamond membranes with thick-
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ness ranging from 100 nm to 400 nm. The thermal conductivity measured on the polycrystalline

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v diamond membrane support a significant grain size effect on the thermal transport.
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PACS numbers: 68.55.-a,73.61.-r,81.05.Uw,81.15.Fg
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Electronic address: [email protected]
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I. INTRODUCTION
In recent years, the constant increase of interest for nanomaterials in thermal physics
(nanophononics) [1–3] and for thermoelectrics [4–6] has motivated the development of very
fine thermal measurement techniques dedicated to small size systems. For instance the
possibility of playing on the geometry of sample or on the nanostructuration of thin films
to reduce the contribution of phonon to the thermal conductivity is a very active subject
of research [2, 7, 8]; the reduced size of the studied objects requires specific experimental
methods. Size effects are studied on the heat conduction (effect on the phonon mean free
path [9], on the dispersion relation [10] or on the transmission coefficient [7, 11]) and to
mention only few inphononic crystals, nanoparticles embedded ina matrix, nanomembranes
or in the presence of rough surfaces etc... Moreover especially for grown thin films, the
thermal properties may strongly depend along which axis they are measured and therefore
their measurements need adapted experimental techniques from high temperature [1] to low
temperature [12].
Since two decades, numerous static or dynamic methods have been developed to mea-
sure the thermal conductivity of new materials like 3ω [13–15], hot wire (Vo¨lklein method)
[16, 17], thermoreflectance [1, 18] or steady state methods [19–22]. The need for precise mea-
surement of very thin films or membranes imposes to work with suspended systems. How-
ever, very few techniques permits such achievement on membrane and nanowire [16, 23–25],
especially when a high sensitivity is necessary. Here we report on a very sensitive dynamic
method based on a mix of the Vo¨lklein and the 3ω methods to measure the in-plane thermal
conductance of membranes. The sensor isconstituted by a thin rectangular suspended mem-
brane with a highly sensitive thermometer lithographied in the center of the membrane. The
thermal gradient is established between the center of the membrane and the frame which
is regulated in temperature. The thermal conductance is deduced from the voltage signal
measured at the third harmonic appearing across the transducer.
II. EXPERIMENTAL
Theexperimentshavebeenperformedusingamixofthe3ω andVo¨lkleinmethods[13,16].
The measurement system consists of a heater-thermometer centered along the long axis of
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a rectangular membrane, which can be easily downscaled. The principle of the method
consists in creating a sinusoidal Joule heating generated by an a.c. electric current flowing
across a transducer. The center of the membrane is thermally isolated from the frame and
hence its temperature is free to increase. The amplitude of the temperature increase or its
dependence on the frequency of the excitation is entirely related to the thermal properties of
the membrane. By measuring the V voltage appearing across the transducer, it is possible
3ω
to deduce the thermal conductivity and the specific heat. The transducer is made out of a
material whose resistance is strongly temperature dependant. It serves as a thermometer
and heater at the same time.
The elaboration of the membranes is detailed on the Fig. 1. The amorphous SiN and
polycrystalline diamond films, which have the advantage to be KOH resistant, are grown
on both side of a silicon substrate by low-pressure chemical vapor deposition (LPCVD) and
micro wave chemical vapor deposition (MWCVD) respectively. The 1 mm long and 150 µm
large membranes are patterned on the rear side by photolithography. After removing the
silicon nitride by SF Reactive Ion Etching, the exposed silicon on the rear side is removed
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in KOH. Finally, rectangular SiN membranes are obtained on the front side. The process is
similar for the diamond membrane. The rear windows are opened using photolithography.
A 100 nm aluminium film protects the diamond outside the patterns. The non protected
diamond is removed by O Reactive Ion Etching and then the silicon is removed by KOH
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etching.
The transducers are patterned on the membranes by regular photolithography. They
consist of niobium nitride and are grown using a dc-pulsed magnetron sputtering from a
high purity (99%-95%) Nb target in a mixture of Ar/N . This type of high sensitivity
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thermometer is described in details elsewhere [26]. Its temperature coefficient of resistance
(TCR) can be tailored over a wide temperature range, from low temperature [27] to high
temperature [28]. Hence, depending on the stoichiometry, the electrical properties of the
NbN can vary a lot. For the SiN measurement, the thermometer has been designed for the
100K-320Ktemperaturerangeandforthediamondmembrane forthe10K-100K.Typically,
the resistance of the thermometer is about 130 kΩ at room temperature with a TCR of 10−2
K−1 and 1MOhm at 70K with a TCR of 0.1K−1. The resistance of the thermometer on
membrane is calibrated using a standard four probe technique between 4 K and 330 K in a
4He cryostat.
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FIG.1: (color online)Schematic oftheSiN(a)anddiamond(c)membranesfabrication. (a)1)The
patterns of the membranes are created by photolithography. The non protected SiN is removed
by SF RIE. 3) The silicon is anisotropically etched in a KOH solution. 4) The thermometers
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are patterned by photolithography. 5) NbN is deposited by reactive sputtering. 6) The resist is
removed. b) In the center of the figure a photograph of the sample is shown: the NbN is grey and
the membrane is yellow(1mm long and 150µm wide).
The ac current induces temperature oscillations of the membrane with an angular fre-
quency 2ω. Consequently the thermometer resistance varies with the same angular fre-
quency. Finally, the measured voltage, due to the thermal oscillations, varies with an angu-
lar frequency 3ω. This V voltage depends on the geometry, the thermal conductivity and
3ω
the specific heat of the membrane. However, the V signal is still present and is much larger
ω
than the V signal by a factor of 103. In the following, we explain how by using a specific
3ω
Wheatstone bridge [29] we strongly reduce the component of the measured voltage at angu-
lar frequency 1ω. The bridge consists of the measured sample (resistance R ), which is the
e
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NbN thermometer on the SiN membrane, the reference thermometer (R ), an adjustable
ref
resistor R and an equivalent non adjustable resistor R = 50KOhm as schematized on the
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Fig. 2. The reference thermometer (or reference transducer) has the same geometry and is
deposited in the same run as the transducer on the membrane. The two resistors R and
v
R are positioned outside the cryogenic system. If R = R and R = R , the electrical
1 e ref 1 v
potential at angular frequency ω is the same in C andD. Consequently, there is no voltage at
the angular frequency of ω between C and D. Since the reference thermometer is not on the
membrane, its temperature remains at T and therefore, its resistance does not change. The
b
elevation of temperature due to self heating of the reference transducer is neglected thanks
to the infinite reservoir of the bulk silicon as compared to the membrane. In that geometry,
the voltage at 1f (V ) has been reduced by a factor of 103. Thus, it is possible to measure
1ω
the V signal between C and D without the 1ω component saturating the dynamic reserve
3ω
of the lock-in amplifier. The two NbN thermometers have practically the same temperature
behavior as they have been deposited simultaneously on the SiN substrate. However, due
to the presence of inhomogeneity in the deposition process, there is a slight difference of
resistance. Thus the R resistor is used to balance the bridge. Thanks to the Wheatstone
v
bridge, the V signal is larger than to the V signal.
3ω 1ω
The heating current is generated by applying an alternative voltage V between A and
ac
B with the oscillation output of the LI 5640 lock-in amplifier. The measured voltage is
frequency filtered (for f 50 kHz) and preamplified by a factor of 100 with a low noise
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preamplifier EPC-1B [30]. EPC-1B is a preamplifier developed at the Institut N´eel with
an input noise around 1 nV/√Hz between 1 Hz and 1 kHz. In the next sections, the V
3ω
component will be presented as a function of the frequency, the amplitude of the excitation
voltage V and as a function of the temperature.
ac
The membrane is installed on a temperature regulated stage and protected by a thermal
shield as schematized on the inset of the Fig. 2. The copper shield, which is maintained at a
temperature close to T , reduces strongly the radiation heat transfer. The thermal gradient
b
between the thermal shield and the sample has been estimated to be much less than 1K,
giving a power of 1Watt per meter square exchanged between the membrane and the shield.
This is equivalent to a parasitic thermal conductance of 10−7W.m−1.K−1. Consequently, any
radiative heat transfer will be neglected in the following. The stage temperature is regulated
with a stability of the order of few milliKelvin. The stage temperature T may be varied
b
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FIG. 2: (color on line) Schematic of the electrical measurement system including HF filter, pream-
plifier and lock-in amplifier. A, B,C and D represent the nodes of the Wheatstone bridge. The
V is measured between C and D. The transducer is referred as R and the reference resistance
3ω e
as R . The inset presents a schematic of the membrane fixed on the temperature regulated stage
ref
covered by the thermal copper shield.
from 4 K to more than 330 K.
III. THERMAL AND ELECTRICAL MODELS OF THE SYSTEM
Themembrane is represented intheFig. 3a. Asthereis asymmetric axiscoming through
the middle of the transducer, the thermal system can be modeled using half the membrane
and half the heating power. For simplification, as the membrane is thin (e 400 nm), we
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assume that the part of the membrane just beyond the NbN transducer is heated like the
thermometer. As the membrane is suspended in vacuum, we assume that the heat can only
diffuse through the membrane toward the silicon substrate which is at constant temperature
T . Thus, in first approximation, we consider a one-dimensional model. The radiative heat
b
loss is neglected as a thermal shield is put between the sample and the calorimeter wall,
as explained in the previous section. Therefore, the system can be modeled as a volume of
′
matter with a total specific heat C and bonded to the thermal bath by the membrane with
a thermal conductivity k. The thermal system is schematized on the Fig. 3 b). The total
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FIG. 3: (color on line)(a)Schematic of the membrane on which the NbN transducer has been
′ ′
deposited. e is the thickness of the membrane and e the thickness of the NbN transducer. C is
the specific heat of the hatched area in red. (b) Schematic of the thermal system.
′
specific heat C take into account both the NbN thermometer and the part of the membrane
′
below the transducer (cf. Fig 3). C can be written as:
b b
′ ′
C = ρ c Le +cρ Le (1)
NbN NbN
2 2
with c the specific heat and ρ the density of the SiN membrane. The temperature is given
by the 1D heat diffusion equation:
∂2T(x,t) 1 ∂T(x,t)
= (2)
∂x2 D ∂t
with D the diffusivity of the membrane. Calculations using a 2D model give approximately
the same results. In order to calculate the solution of Eq. 2, we need initial and boundary
conditions. Therefore, we assume that at t=0, the temperature of the membrane is T since
b
the transducer is not heated. Moreover, we assume that the membrane edge is always at
T=T . Thus, the initial and boundary conditions can be written as: T(x,t = 0) = T ,
b b
T(x = 0,t) = T . Moreover, the total dissipated power P(t) is used to heat both the
b
thermometer and the part of the membrane under the thermometer, and the rest of the
membrane: C′(T)∂T(x,t) =P(t)-eLk∂T .
∂t x=ℓ ∂xx=ℓ
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The general solution of Eq. 2 is:
P sh[ω′(1+j)x]ej2ωt
0
T(x,t) = (3)
(1+j)Skω′ch[ω′(1+j)ℓ]+j2C′ωsh[w′(1+j)ℓ]
with ω′ = ω, S=eL, P =RI02 and I = I sin(ωt). We can also write Eq. 3 using
D 0 4 0
q
exponential notation:
T(x,t) = P0 sin2(ω′x)+sh2(ω′x) 1/2ej2ωt+ϕ (4)
1/2
D
0 h i
1/2
with ϕ the phase and D the module of the denominator of Eq. 3. After development
0
in Taylor expansion in first order in ω, the expression of the temperature module T (ℓ) can
m
be written as followed:
P
0
T (ℓ) = (5)
m K 1+ω2 4τ2 + 2ℓ4 + 4τℓ2 1/2
p 3D2 3D
with K =kS, τ=C′ and D the thermhal diffu(cid:16)sivity. (cid:17)i
p ℓ Kp
Once the temperature variation is known, we can calculate the V voltage between C
3ω
and D. As the distance between the thermometer and the resistances R and R is not
v 1
negligible, we assume that there is line electric capacities (C ) in parallel, as schematized
l
on the Fig. 4. Moreover, we assume that the two thermometers present also an electrical
′ ′
capacity: C = 2C + C for the reference and C = 2C + C for the sample. Besides, we
3 l 3 4 l 4
consider the transducer on membrane as a 3ω voltage generator (U ) since it delivers a
3ω
current at angular frequency 3ω (see Fig 4). In addition, as the output impedance of the
lock-in amplifier is high, we suppose that the 3ω current remains in the Wheatstone bridge
system. Following the schemes given in Fig. 4, the V and V voltages can be written as
1ω 3ω
follows: the module of V :
1ω
V ǫ2 +R2R2 ω2(R C′ R C′)2 1/2
V (ω) = ac e ref 1 4 − v 3 (6)
| 1ω | (R +R )2 +(Rh R C′ω)2 1/2 (R +R )2 +(iR R C′ω)2 1/2
v e v e 3 ref 1 1 ref 4
h i h i
with ǫ = R R R R and the phase ϕ given by ϕ = ϕ ϕ ϕ where :
ref 1 e v 3 4 5
− − −
′ ′ ′ ′
R R (R C R C )ω R R C ω R R C ω
tgϕ = ref e 1 4 − v 3 tgϕ = v ref 3 tgϕ = 1 e 4 (7)
3 4 5
R R R R R +R R +R
1 ref e v ref v e 1
−
and the thermal voltage generated U (see Fig 4) is given by:
3ω
U (ω) = Z αT(ℓ,t)I (8)
3ω e
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FIG. 4: (color on line) Electrical schematic of the Wheatstone bridge. Z is the impedance of the
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lock-in amplifier input. The transducer is modeled as generating a voltage U at 3ω.
3ω
′
with Z = R /(1+jR C ω) and I the current coming through the impedance Z .
e e e 4 e
Using the relations between the currents in the Wheatstone bridge, we can obtain the
following equation:
′
V 1+jR C ω
ac e 4
I = (9)
(cid:16) (cid:17)′
R +R +jR R C ω
1 e e 1 4
(cid:16) (cid:17)
Then, the general expression of V , between C and D, can be written as follows:
3ω
R T ejϕV α(R +R )(1+jR C ω)
e m ac 1 v ref 3
V (ω) = (10)
3ω ′
R +R +jR R C ω [R +R +R +jR (R +R )(2C +C )ω]
1 e 1 e 4 1 v ref ref 1 v l 3
(cid:16) (cid:17)
with V the voltage put on the Wheatstone bridge (between A and B),ϕ the thermal
ac
phase, C the line capacity (cf. Fig.), C and C being the electric capacity of the reference
l 3 4
thermometer and of the thermometer on the membrane respectively.
The module of V is given by:
3ω
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R T Vrmsα(R +R ) 1+(R C ω)2 1/2
Vrms(ω) = e m ac 1 v ref 3
| 3ω | (R +R )2 + R R C′ω 2 1/2 (R +R +Rh )2 +[R (Ri +R )(2C +C )ω]2 1/2
1 e 1 e 4 1 v ref ref 1 v l 3
(cid:20) (cid:16) (cid:17) (cid:21) n (11) o
and the phase by:
′
R R C ω R ω(R +R )(2C +C )
ϕ (ω) = ϕ+arctan(R C ω) arctan e 1 4 arctan ref 1 v l 3
V3ω ref 3 − R1 +Re !− " R1 +Rv +Rref #
(12)
All the fits on V and V of this work have been realized with these equations. The
1ω 3ω
Fits of our results show that, for frequency below 1 kHz, the electric capacities of the
thermometers are negligible as compared to the thermal effects. Thus, in order to give a
′
clearer and more physical description of the method, we can consider that C =C =C =0.
3 4 l
Then the expression of V becomes:
3ω
α(Vrms)3(R +R )R2
Vrms(ω) = ac 1 v e (13)
| 3ω | 2K (R +R )3(R +R +R ) 1+ω2 4τ2 + 2l4 + 4τl2 1/2
p 1 e 1 e v 3D2 3D
h (cid:16) (cid:17)i
At low frequency, the ω term becomes negligible and the expression of V can be written
3ω
in its simpler form as:
α(Vrms)3(R +R )R2
Vrms(ω) = ac 1 v e (14)
| 3ω | 2K (R +R )3(R +R +R )
p 1 e 1 v ref
Thus, at low frequency, V is constant and depends on the thermal conductivity k of the
3ω
membrane. After the calibration of R and R ,R and R being fixed, a measure of V at
e ref 1 v 3ω
low frequency allows the calculation of the thermal conductance K using Eq. 14.
p
IV. RESULTS
The method has been checked using two different measurements. According to equation
14, the V signal depends on the cube of the Wheatstone bridge voltage V . Thus, the V
3ω ac 3ω
signal has been measured at different temperatures to check this behaviour. As an example,
the 100 K measurement can be seen on the Fig.5. The linear fit gives a slope very close to
3 which confirms the cubic behaviour of the V signal versus the applied voltage V .
3ω ac
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